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Mersenne primes (of form 2^ p − 1 where p is a prime) In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The ...
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and its Linux port MPrime. Scott Kurowski wrote the back end PrimeNet server to demonstrate volunteer ...
The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his Cogitata Physico-Mathematica (1644; see e.g. Dickson 1919) that the numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n ≤ 257. The first seven entries of his list ( for n = 2, 3, 5, 7 ...
As of 2018, there are 51 known Mersenne primes. The 13th, 14th, and 51st have respectively 157, 183, and 24,862,048 digits. As of 2018, this class of prime numbers also contains the largest known prime: M 82589933, the 51st known Mersenne prime.
The last seventeen record primes were Mersenne primes. [3] [4] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2 k − 1 is simply k ones. [5] Finding larger prime numbers is popularly understood to permit stronger encryption, yet this is incorrect. [6] [7]
Perfect number. In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
a ( n) = 2^ (2^prime ( n) − 1) − 1. A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a double Mersenne number can be prime only if Mp is itself a Mersenne prime. For the first values of p for which Mp is prime, is known to ...